Ch3 EconS301 revised -...

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Consumer Preference and The Concept Of Utility

Representation of Preferences Bundle/basket‐ a combination of goods and services that an individual might consume.Example: Bundle A = (60, 30) contains 60 units of food and 30 units of clothing.

Consumer preferences: an indication of how a consumer would rank the desirability of baskets.

Consumer Preference‐Assumptions

1) Preferences are complete:

Ability to compare, either A >B or B >A, or A~B (“A is indifferent to B”)

Preferences must to state that either the consumer prefers one basket to another or he is indifferent.

2) Preferences are transitive:

For three bundles A, B, and C, if A >B and B >C, then A >C.

If we allow for non‐transitive preferences, we can show that they must cycle, thus being prove to exploitation.

Consumer Preference ‐ Assumptions

Example of non‐transitive preferences:

“I would rather have an orange than a banana. And I would rather have a banana than an apple. However, (puzzlingly) I would rather have an apple than an orange.”

1st 2nd 3rd

Orange > Banana Banana > Apple Apple > OrangeOrange ‐ $1 Banana ‐ $1 Apple ‐ $1

But now the intransitive consumer is at the original position (owning an orange), and we can start the money pump again.

Individuals with intransitive preferences would then be wiped out of the market.

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Consumer Preferences‐Assumptions 3) More is better (also referred to as “nonsatiation”)

A consumer should prefer two hot dogs to one, and three hot dogs to two.

This assumption helps rank some of the bundles in the previous figure.

A > E > H

A > B > J And A > G > D

But it doesn’t help us rank other bundles:

G > E ? We don’t know.

Ranking Systems Ordinal Ranking

allows us to gather information about the order in which a consumer ranks a bundle. We can know that a consumer prefers basket A to B but not how much more we prefers A to B.

We only need ordinal ranking in order to develop all our analysis of consumer theory.

Cardinal Ranking allows us to answer the intensity at which a consumer prefers one bundle to another. For example, I like basket A twice as much as basket B. It is usually hard for consumers to articulate their intensity.

Utility Functions It measures the level of satisfaction from consuming different bundles. It represents the consumer’s preferences. The unit of measurement is Utils.

We can think of Utils as a measure of a consumer’s happiness in consuming a good, or a bundle of goods.

Utility Functions – Example

Example: A utility function for a single good.

Does it meet the above 3 assumptions?

yyu y 2/1

)(

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Utility Functions ‐ Example Completeness

For any two bundles A and B, we must have that either u(A) ≥ u(B), or u(A) ≤ u(B), or u(A) = u(B), which indicates that either A>B, A<B, or A~B. Hence, Completeness is satisfied since for any two bundles x and z, either

u(x) = √x> √z = u(z), oru(x) = √x< √z = u(z), or

u(x) = √x = √z = u(z).

Example: A ya = $4 u(4) = √4 = 2B yb = $9 u(9) = √9 = 3

Utility Functions – Example Transitivity

For example, for three bundles

C=1, B=4, A=5, then u(C) = (1)1/2 = 1

u(B) = (4)1/2 = 2

u(A) = (5)1/2 =2.23

Thus if u(A) > u(B) and u(B) > u(C), then we have that u(A) > u(C).

Therefore the preferences are transitive because:

if A >B and B >C, then A >C.

Note that this property holds for any three bundles (numbers, since we are talking about quantities of a single good) a, b, and c. In particular, if √a > √b and √b > √c then it must be that √a > √c.

AB

C

541

u(A) 5

u(B) 2

u(C) 1

u(x) x

u(A) u(B)

and

u(B) u(C)

implies;

u(A) u(C)

Utility Functions ‐ Example Non‐satiation (more is better)

Indeed it is satisfied since u(B) > u(A) for any two bundles in which B is larger than A

Example:

A = 1 unit B = 2 units

U (1) = √1 = 1 u (2) = √2 > 1.

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Utility Functions Marginal Utility‐ the rate at which utility changes as consumption of a good increases.

Interpretation: how much better off we would be if we received one more unit of good y?

In mathematical terms, it is the first derivative of the Utility Function with respect to good y.

That is, the slope of the utility function, i.e., its rate of growth as we consume more units of the good.

y

yu

y

uMU y

)(

Utility Functions ‐ Example

Utility Functions‐Example

Mathematical Example

As illustrated in the figure u(y) = √y is increasing in y, but its marginal utility, is decreasing in y.

1 11

2 2

( ) ,

1 1

2 2

1 1

2

1

2

If u y y

uthen y

y

y

y

y

yy

u

2

1

Utility Functions

Three Important Points when drawing Total and Marginal Utility Graphs:

1) Total Utility and marginal utility cannot be plotted on the same graph, because the variable in the vertical axis differs on each graph.

2) The marginal utility function is the slope of the total utility function.

3) The relationship between total and marginal utility holds for other measures in economics that will be discussed later on in the semester (such as total and marginal product for a firm).

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Utility Functions

Diminishing Marginal Utility: additional units add less utility (∆U smaller and smaller)

Indeed, marginal utility

is decreasing in y.

Nonetheless, we will assume that ∆U > 0 always.

That is, more is always better, although the last unit consumed is not as good as the first unit consumed (Think about your first glass of water on a hot day and your fourth glass of water).

Utility Functions Utility Functions with only one good are very restrictive.

We seek to generalize our analysis to more than one product.

Utility functions with more than one good, e.g,

xyyxu ),(

Utility Functions Utility functions with more than one good, e.g,

Point A, (x,y)=(2,8) provides the same utility as Point B (x,y)=(4,4), since they both reach the same height.

xyyxu ),( Utility Functions Utility functions with more than one good.

Mathematical Example:

Let us now find the marginal utilities

xyy

yxUMU

yxx

yxUMU

y

x

2

1),(2

1),(

Amount of y

is fixed

Amount of x is fixed

yxxyyxU *),(

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Utility Functions‐Exercise Learning by doing 3.1

U(x,y) =

Satisfies more is better?

Satisfies diminishing marginal utility?

More is better must be checked in two different ways: Increase in x increase in U

Increase in y increase in U

Alternatively, we can check if MUx > 0 and MUy > 0 for all x,y > 0. This property is satisfied since:

MUx = is positive for any amount of good x and y

Similarly,

MUy = is positive for any amount of good x and y

Utility Functions – Exercise Diminishing marginal utility just tells us that the additional utility

we get from consuming further amounts of goods is smaller and smaller.

The additional utility we get from consuming additional goods is MUx

and MUy. So we need to show:

MUx is decreasing in x (it turns out to be true)

MUy is decreasing in y (it turns out to be true)

Mathematically, in order to see whether MUx is decreasing in x, we need to check the second derivative of the utility function with respect to x. That is:

MUxx < 0 and MUyy < 0 for all x,y > 0

04

2

1MU

2/3

yy

y

x

yxy

for all x and y04

2

1MU

2/3

xx

x

y

xyx

for all x and y

Query #1 Consider the utility function U = min ( ).

Does good x satisfy the diminishing marginal utility property?

A. Yes

B. No

C. Depending on the number of units of good y that the individual consumes.

3x 2 7y 3

Query #1‐ Answer

• MUx is which increases in x

• Indeed,

• Hence, u(x,y) = does not satisfy the diminishing marginal utility property fro good x.

• As a practice, show that u(x,y) doesn’t satisfy it for good y either.

dU

dx 6x

dMUx

dx

d(6x)

dx 6 0

3x 2 7y 3

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What about a different utility function, such as u(x,y)= ?

Then, MUx = , which is still increasing in x.

Therefore, diminishing marginal utility doesn’t hold in this case either.

3x 2 7y 3

6x5y 3

Indifference Curves Indifference Curves: curves connecting consumption bundles that yield the same level of utility.

Indifference Curves For utility function u(x, y) = bundles A, B and C yield the same utility level (and thus lie on the same indifference curve).

Graphically, all three bundles help a consumer reach the same height in his “utility mountain” (i.e., the 3D representation of his utility function we depicted a few slides ago).

In particular,

u(8,2) = 8 ∗ 2 16 4 for bundle A,

u(4,4) = 4 ∗ 4 16 4 for bundle B,

u(2,8) = 2 ∗ 8 16 4 for bundle C.

Indifference Curves – Properties 4 properties of indifference curves (1 of 4):

1. When the consumer likes both goods (MUx>0, MUy>0), indifference curves are negatively sloped

Indeed, by the “move is better” assumption (i.e., MUx > 0 and MUy > 0), points to the northeast of A must be strictly better, while points to the southwest of A must be strictly worse. Hence, none of the shaded areas can lie on the same indifference curve as Point A

As a consequence, the indifference curve of bundles yielding the same utility as bundle A must lie on the unshaded areas. Thus, indifference curves must be negatively sloped.

MUx > 0

MUy > 0

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1. Indifference curves cannot intersect.

2. If they did (as in the figure) then R>Q (implying that U2 > U1), but S>T (implying that U1 > U2), which leads to a contradiction.


1. Every consumption bundle lies on one and only one IC. 2. This property follows from the previous. (We cannot have

points like A, which lies on two different I.C.s and thus would have two different utility values)


1. ICs are not thick. 2. Otherwise, bundle B would yield the same utility level as A, which

cannot be since B is to the northeast, containing more units of both x and y, thus it must be strictly preferred to A (by the “more is better” assumption).

Indifference Curves Marginal Rate of Substitution: The rate at which a consumer will give up one good (y) to get one additional units of another good (x), keeping his or her utility level constant.

Where is this expression of MRSx,y coming from?

(Proof on the next slide)

y

xxy MU

MUICofslopeMRS

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MRS of u(x,y) First, note that we need to totally differentiate with respect to x and

y (indeed, when moving rightward along an indifference curve, we are simultaneously varying x and y, i.e., lowering y in order to get more of x).

Totally, differentiating, we obtain

Since we move along the same indifference utility is unaffected, i.e.,

Rearranging,

u MUxx MUyy

0 MUxx MUyy

ICofslope

ICofslopex

y

MU

MU

yMUxMU

MRSxy

y

x

yx

∆u = ∂u/∂x ∙ ∆x + ∂u/∂y ∙ ∆ y

u 0

MRS of u(x,y) Diminishing MRS:

Initially you are willing to give up many glasses of lemonade (y) for an additional hamburger (x). However, when you have few glasses of lemonade and many hamburgers you will not be willing to give up as many (or none) glasses of lemonade in order to get an additional hamburger.

That is, IC’s are bowed in towards the origin.

Decreasing slope of I.C.

Decreasing MRSx,y

x

y

MRS of u(x,y)

The same increase in one hamburger (good x) leads the consumer to give up many glasses of lemonade when he is at bundle A (consuming a lot of lemonade), but to give up only a few glasses of lemonade when he is at point C (when he has few glasses of lemonade).

u(x,y) = u

A

B

C

D

y

x

y

y

1 unit 1 unit

Indifference Curve ‐ Application Demand for attributes in cars

where x is gas mileage and y is horse power

MRSx,y in this context represents how a typical consumer is willing to forgo horsepower (y) in order to get one more unit of gas mileage (x).

In 1969, MRSx,y=3.79, while in 1986, such MRSx,y dropped to 0.71 showing that people are less willing to give up horsepower (y) in order to get additional units of miles per gallon (x)

U(x,y) xy

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On average, consumers were willing to forgo 3.79 horsepower to get an additional one mile per gallon in 1969, but are only willing to forgo .71 horsepower to get an additional one mile per gallon in 1986.

Indifference Curves ‐ Exercise Learning by doing 3.3

a) Draw the IC corresponding to utility level of U1 = 128xy = 128 in many of its combinations and connect them

For example, G: x = 8 , y = 16 H: x = 16, y = 8I: x = 32, y = 4

Generally, the trick is x ∙ y = 128 y = 128/x y = 128/8 = 16 Point G

y = 128/16 = 8 Point Hy = 128/32 = 4 Point I

U (x, y ) xy

MU x U (x, y )

x y

MU y U (x, y )

y x

Indifference Curves – Exerciseb) Does I.C. intersect either axis?

No, if IC intersected x‐axis, then at the crossing point, y = 0, utility level would be U1 = 0 ≠ 128 , Similarly, if IC intersected y‐axis, then at the crossing point, x = 0, utility level would be U1 = 0 ≠ 128

, ∗

Indifference Curves – Exercisec) Does IC indicate that MRSx,y is diminishing?

Yes, since IC is bowed in toward the origin.

Note that:

so as we move from left to the right (∆x), we get smaller ratios, smaller MRSx,y (diminishing MRS), i.e., the indifference curve becomes flatter.

, ∗

x

y

MU

MU

y

xxyMRS

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Query #2Consider the utility function U =5x +3y, which has the following marginal utilities MUx =5 and MUy = 3. The indifference curves for this utility function:

A. Will have a diminishing marginal rate of substitution of x for y as x increases.

B. Will have a constant marginal rate of substitution of x for y as x increases.

C. Will have an increasing marginal rate of substitution of x for y as x increases.

Query #2 ‐ Answer Answer B

Taking utility function U (x,y)=5x+3y , and finding the MUx =5 and MUy = 3, we obtain the MRS between goods x and y is 5/3.

Hence, the MRS is constant in x

Hence, answer B is correct.

Example of an Increasing MRSx,y U(x,y)=Ax2+By2

In this case, MUx=2Ax and MUy=2By

Hence,

Unlike in previous examples, these indifference curves are bowed away from the origin

MRSx,y MUx

MUy

2Ax

2Ay

Ax

By, which increases in x

Special Utility Functions‐Perfect Substitutes

Perfect Substitutes: Aquafina versus Dasani bottled water; or Kingston versus Scandisk memory sticks. Close substitutes: butter versus margarine; or coffee versus black tea.

MRSB,M = MRSM,B = constant

Example: Butter (B) and Margarine (M) U = aB + bM, then MUB = a and MUM = b Hence, MRSB,M =

Then , slope of IC = (constant, i.e., independent in x)

Note that if u = aB + aM, so MUb = a and MUm = a,Then MRSb,m = a/a =1

MUb

MUm

a

b

a

b

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Special Utility Functions ‐ Perfect Substitutes

Example: Pancakes and Waffles U = P + 2W, which implies MUp = 1 and MUW = 2.

A waffle gives him double utility of a pancake.

The consumer is always willing to trade 2 pancakes for 1 waffle.

slope of IC = ‐.5 The slope of the indifference curve

is constant

Hence, the MRS is also constant

MRSp,w MUp

MUw

1

2

(It doesn’t change in the amount of pancakes)

I.C. are straight lines

Special Utility Functions ‐ Perfect Complements Perfect Complements: left

shoe and right shoe (consumer wants them in fixed proportions), cars and gasoline, or two scoops of peanut butter to one scoop of jelly for the perfect PB&J sandwich.

U = 10×min(R,L)

At G, we have that R=2 and L=2, then U2=10*min(2,2) = 10*2=20

At H, we have that R=3 and L=2, then U2=10*min(3,2) = 10*2=20

Consumption choices should be allocated at the kink (Why spend money on additional units of R if they won’t increase your utility at all?)

Special Utility Functions ‐ Perfect Complements

More generally, utility functions for goods that are perfect complements look like:

u (x1, x2) = A ∙ min (a x1, b x2),

where A, a, b > 0 are all positive constants.

Special Utility Functions‐Perfect Complements

Slope of IC (MRS) is:

Zero at the flat segment of the IC

Infinity at the vertical segment of the IC

Undefined at the kink of the IC (infinitely many slopes)

Slope = ‐∞

Slope = 0

Slope at the kink is undefined

μ1

L

R

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Query #3Identify the truthfulness of the following statements:

I. Diminishing marginal utility and increasing total utility are incompatible with each other.

II. When Marginal utility is negative, total utility is decreasing.

a) Both I and II are true

b) Both I and II are false

c) I is true; II is false

d) I is false; II is true

Query #3 – Answer Answer D

Statement I is false, because total utility can increase and yet experience diminishing marginal utility, i.e., in the increasing, but concave, portion of the total utility, additional increments of utility become smaller and smaller (diminishing MU), but total utility is still increasing.

Statement II is true, since for marginal utility to become negative, it must be that additional consumption actually reduces your total utility, i.e., total utility is decreasing.

For a graphical representation of total and marginal utility, see slide 13 of this chapter.

Special Utility Functions – Cobb Douglas Cobb‐Douglas Utility function

and U = xy are examples of Cobb‐Douglas utility functions

Generally, Cobb‐Douglas utility functions look like

U(x,y) = Axαyβ where parameters A, α, β > 0 are all positive constants.

if α = β = ½, then U = Ax1/2y1/2 = A(xy)1/2

if α = β = 1 and A =1, then U = xy

U(x,y) xy

Special Utility Functions – Cobb Douglas The Douglas utility function u(x,y)=Axα yβ satisfies 3 properties:

1) MUx > 0 and MUy > 0. That is, “more is better” property holds

MU xU ( x , y )

x Ax 1y 0 for any x

MU yU ( x , y )

y Ax y 1 0 for any y

and y

and x

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Special Utility Functions ‐ Cobb‐Douglas2. Since MUx > 0 and MUy > 0, then IC is downward slopping (recall property 1 of ICs)

[what about the case in which the consumer dislikes some goods, MUx < 0? You will see that in the Review Session]

3.Diminishing MRSx,y:

Then movements from left to right (∆x) will shrink this ratio,

and reduce MRSx,y Diminishing MRSx,y (Flatter I.C.)

11 ( 1)

, 1* *

1* * *

x y

A x yMRS x y

A x y

yy

x x

Special Utility Functions ‐ Quasilinear Utility

Quasi‐linear: Used in economic applications for situations in which the amount of a commodity (such as toothpaste or garlic) doesn’t change very much with income.

ICs are parallel displacements to each other

Same slope, same MRS between goods, even if income changes.

e.g, toothpastegarlic etc.


General Form of Quasilinear utility: U(x,y) = ν(x) + by

(linear in y, but not linear in x)

where ν(x) is a function that increases in x,

such as ν(x) = (x)1/2 or ν(x) = x2.

and where b > 0 is a positive constant.

If we find the MRSx,y,, we obtain

That is, for a given value of x, the MRSx,y is constant in y.

b

xv

MU

MU

y

x )(MRS yx,


General Form of Quasilinear utility: U(x,y) = ν(x) + by

(Example: U(x,y)= x2 +3y. In this case, MRSx,y is

For a given x=5, MRSx,y =10/3, which is constant in y (as depicted in the previous figure).

3

2MRS yx,

x

MU

MU

y

x

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Pet Rock Fad as an example of quasilinear utility.In case you didn’t remember what Pet Rocks are… surprisingly, they were famous a few years ago.


Pet Rock Fad Application: observe the differences in the graphs.

Pet Rock Fad: all additional income is spent on pet rocks.

The fad is almost over: additional income is spent on both goods.

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